Method for evaluating and preventing creep damage to conductivity of hydraulic fracture in gas reservoirs

Abstract

The present invention discloses a method for evaluating and preventing creep damage to conductivity of hydraulic fracture in gas reservoirs, comprising: (1) selecting a rock sample of target reservoir for creep experiment, and plotting ε-t curve of the rock sample during creep; (2) fitting the fractional Kelvin model with the ε-t curve of the rock sample during creep; (3) calculating the conductivity and permeability of hydraulic fracture considering creep damage; (4) numerically solving the productivity model, calculating the cumulative gas production of the gas well produced up to time t, and calculating the creep damage rate for cumulative production of the gas well; (5) repeating Steps (3) to (4), calculating the creep damage rate for cumulative production for the cases of hydraulic fracture sanding concentration N of 5 kg/m.sup.2, 7.5 kg/m.sup.2, 10 kg/m.sup.2, 12.5 kg/m.sup.2 and 15 kg/m.sup.2 respectively, plotting the creep damage chart of cumulative production.

Claims

1. A method for evaluating and preventing a creep damage to a conductivity of a hydraulic fracture in gas reservoirs, comprising the following steps: (1) selecting a rock sample of a target reservoir for a creep experiment, and plotting a strain ε versus time t curve of the rock sample during the creep experiment; (2) fitting a fractional Kelvin model with the strain ε versus time t curve of the rock sample during the creep experiment, and a fitting equation of the fractional Kelvin model is as follows: $\begin{array}{cc}\mathrm{.Math.}=\frac{{\sigma }_z}{E_v}\left[1-E_{\alpha }\left(-\frac{t^{\alpha }}{\tau }\right)\right];& \left(1\right)\\ E_{\alpha }\left(-\frac{t^{\alpha }}{\tau }\right)=\underset{n=0}{\overset{\infty }{.Math.}}\frac{{\left(-\frac{t^{\alpha }}{\tau }\right)}^n}{\Gamma \left(\alpha n+1\right)};& \left(2\right)\\ \Gamma \left(\alpha n+1\right)={\int }_0^{\infty }{T}^{\alpha n+1-1}{e}^{-T}\mathrm{dT};& \left(3\right)\end{array}$ where, ε—a strain in the rock sample during the creep experiment, dimensionless; E.sub.v—a viscous modulus of the rock sample, in MPa; α—a derivative order, dimensionless; t—a time, in s; τ—a fractional relaxation time, in s.sup.α; σ.sub.z—an axial differential pressure of the rock sample, in MPa; n—a set of natural numbers, 0, 1, 2, 3 . . . ; T—an integral variable, taken as [0, ∞]; fitting them with the strain ε versus time t curve to obtain the viscous modulus E.sub.v, the fractional relaxation time τ, and the derivative order α; (3) substituting a hydraulic fracture width D, the viscous modulus E.sub.v of the rock sample, the fractional relaxation time τ, and the derivative order α into the following equations to obtain a hydraulic fracture conductivity F.sub.RCD and a hydraulic fracture permeability K.sub.FR considering the creep damage: $\begin{array}{cc}\{\begin{array}{c}{F}_{RCD}=\frac{74.8f_1\left(f_1-0.74\right)\left({\mathrm{Df}}_1-{\mathrm{df}}_2\right)}{0.5f_1^2+1}{d}^2\\ f_1=1-2.08{\left(\hat{p}\frac{1-v_1^2}{E_1}\right)}^{\frac{2}{3}}\\ f_2=2.08{\left(\hat{p}\frac{1-v_1^2}{E_1}+\hat{p}\frac{1-v_2^2}{{\hat{E}}_2}\right)}^{\frac{2}{3}}-2.08{\left(\hat{p}\frac{1-v_1^2}{E_1}\right)}^{\frac{2}{3}}\\ \frac{1}{{\hat{E}}_2}=\frac{1}{E_2}+\frac{1}{E_v}\frac{1}{1+\tau s^{\alpha }}\end{array};& \left(5\right)\\ \hat{p}={\int }_0^{\infty }p.Math.e^{-st}\mathrm{dt};& \left(6\right)\\ K_{FR}=\frac{F_{RCD}}{D};& \left(7\right)\end{array}$ where, d—a diameter of a proppant, in mm; v.sub.1—a Poisson's ratio of the proppant; E.sub.1—an elastic modulus of the proppant, in MPa; v.sub.2—a Poisson's ratio of the rock sample of the target reservoir; Ê.sub.2—an elastic modulus of the rock sample of the target reservoir in Laplace space, in MPa; E.sub.2—an elastic modulus of the rock sample of the target reservoir, in MPa; {circumflex over (p)}—a pressure at any position in the hydraulic fracture in Laplace space, in MPa; p—a pressure at any position in the hydraulic fracture, in MPa; s—a Laplace variable, in s.sup.−1; (4) calculating a creep damage rate for cumulative production of a gas well with the following steps: a gas well productivity mode is established, in a x-y rectangular coordinate system, a reservoir length and a reservoir width are denoted as L.sub.x and L.sub.y respectively; L.sub.x is divided into n.sub.i sections in the x direction and L.sub.y is divided into n.sub.j sections in the y direction, so the target reservoir is divided into an n.sub.i×n.sub.j matrix grid, with (i,j) representing a number of each grid cell, and x.sub.i,j and y.sub.i,j representing a length and a width of each grid cell respectively; the hydraulic fracture is divided by the matrix grid into n.sub.l hydraulic fracture segments; the length of the hydraulic fracture segment in a first section is ξ.sub.l; moreover, a local coordinate system of the hydraulic fracture is established, which defines ξ as a fracture direction in the local coordinate system, then: a) differential equations of two-phase seepage of the hydraulic fracture and the matrix are: $\begin{array}{cc}\frac{\partial }{\partial \xi }\left(\beta \frac{K_F{K}_{\mathrm{Frw}}}{{\mu }_w{B}_w}\frac{\partial P_F}{\partial \xi }\right)+\frac{q_{Fw}}{V_F}+\frac{\beta K_m{K}_{mrw}}{{\mu }_w{B}_w}\left(P_{mw}-P_F\right)=\frac{\partial }{\partial t}\left(\frac{{\varphi }_F{S}_{\mathrm{Fw}}}{B_w}\right);& \left(8\right)\\ \frac{\partial }{\partial \xi }\left(\beta \frac{K_F{K}_{Frg}}{{\mu }_g{B}_g}\frac{\partial P_F}{\partial \xi }\right)+\frac{q_{Fg}}{V_F}+\frac{\beta K_m{K}_{\mathrm{mrg}}}{{\mu }_g{B}_g}\left(P_{mg}-P_F\right)=\frac{\partial }{\partial t}\left(\frac{{\varphi }_F{S}_{Fg}}{B_g}\right);& \left(9\right)\\ \nabla \left(\beta \frac{K_m{K}_{\mathrm{mrw}}}{{\mu }_w{B}_w}\nabla P_{mw}\right)-\frac{\beta K_m{K}_{\mathrm{mrw}}}{{\mu }_w{B}_w}\left(P_{mw}-P_F\right){\delta }_m=\frac{\partial }{\partial t}\left(\frac{{\varphi }_m{S}_{\mathrm{mw}}}{B_w}\right);& \left(10\right)\\ \nabla \left(\beta \frac{K_m{K}_{mrg}}{{\mu }_g{B}_g}\nabla P_{mg}\right)-\frac{\beta K_m{K}_{\mathrm{mrg}}}{{\mu }_g{B}_g}\left(P_{mg}-P_F\right){\delta }_m=\frac{\partial }{\partial t}\left(\frac{{\varphi }_m{S}_{mg}}{B_g}\right);& \left(11\right)\\ S_{mg}=1-S_{mw};& \left(12\right)\\ S_{Fg}=1-S_{Fw};& \left(13\right)\\ P_{mc}=P_{mg}-P_{mw};& \left(14\right)\end{array}$ where, K.sub.Frw and K.sub.Frg—a relative permeability of water and a relative permeability of gas in the hydraulic fracture, dimensionless; K.sub.F—a Hydraulic fracture permeability, D; if considering the creep damage, K.sub.F=K.sub.FR; if not considering the creep damage, K.sub.F=K.sub.F0, and K.sub.R) is an initial permeability of the hydraulic fracture; V.sub.F—a volume of the hydraulic fracture segment, in m.sup.3; q.sub.Fw, q.sub.Fg—a source sink terms of water and a source sink term of gas in the hydraulic fracture, in m.sup.3/s; S.sub.Fw, S.sub.Fg—a water saturation and a gas saturation in the hydraulic fracture, dimensionless; φ.sub.F, φ.sub.m—a porosity of the hydraulic fracture and a porosity of the matrix, dimensionless; P.sub.F—a water pressure and a gas pressure of the hydraulic fracture, in MPa; μ.sub.w and μ.sub.g—a viscosity of water and a viscosity of gas, in mPa.Math.s; B.sub.W and B.sub.g—a volume coefficient of water and a volume coefficient of gas, in m.sup.3/m.sup.3; K.sub.m—a permeability of matrix, D; P.sub.mw and P.sub.mg—a pressure of water and a pressure gas in matrix, in MPa; P.sub.mc—a capillary pressure in the matrix, in MPa; K.sub.mrw and K.sub.mrg—a relative permeability of water and a relative permeability of gas in the matrix, dimensionless; S.sub.mw and S.sub.mg—a water saturation and a gas saturation in the matrix, dimensionless; β—an unit conversion factor, taken as β=0.001; t—a time, in s; ξ—a fracture direction in the local coordinate system of the hydraulic fracture; δ.sub.m—an equation parameter, δ.sub.m=1 if there is the hydraulic fracture passing through the matrix grid, or δ.sub.m=0 if there is no hydraulic fracture passing through; b) initial conditions include a distribution of the initial pressure and the initial saturation: $\begin{array}{cc}\{\begin{array}{c}{P_F\left(x,y,t\right).Math.}_{t=0}=P_{F0}\left(x,y\right)\\ {P_{mg}\left(x,y,t\right).Math.}_{t=0}=P_{mg0}\left(x,y\right)\end{array};& \left(15\right)\end{array}$ where, P.sub.F0(x,y)—a distribution of the initial gas pressure in the hydraulic fracture, in MPa; P.sub.mg0(x,y)—a distribution of the initial gas pressure in the matrix, in MPa; x, y—Horizontal and vertical coordinates respectively in the rectangular coordinate system; $\begin{array}{cc}\{\begin{array}{c}{S}_{Fw}\left(x,y,t\right){.Math.}_{t=0}=S_{Fw0}\left(x,y\right)\\ {S_{mw}\left(x,y,t\right).Math.}_{t=0}=S_{mw0}\left(x,y\right)\end{array};& \left(16\right)\end{array}$ where, S.sub.Fw0(x,y)—a distribution of initial water saturation in the hydraulic fracture, dimensionless; S.sub.mw0(x,y)—a distribution of initial water saturation in the matrix, dimensionless; c) inner boundary conditions are: $\begin{array}{cc}\{\begin{array}{c}{P}_F\left(x_w,y_w,t\right)=P_{\mathrm{wf}}\left(t\right)\\ P_m\left(x_w,y_w,t\right)=P_{\mathrm{wf}}\left(t\right)\end{array};& \left(17\right)\end{array}$ where, x.sub.W and y.sub.w—horizontal and vertical coordinates of the grid cell where the gas well is located, in m; P.sub.wf—a bottom hole pressure, in MPa. d) outer boundary conditions are: $\begin{array}{cc}\{\begin{array}{c}{\frac{\partial P_m}{\partial x}.Math.}_{x=0,L_x}=0\\ {\frac{\partial P_m}{\partial y}.Math.}_{y=0,L_y}=0\end{array};& \left(18\right)\end{array}$ where, L.sub.x, L.sub.y—a reservoir length and a reservoir width, respectively, in m; the productivity model is numerically solved to obtain the gas saturation S.sub.Fg in the hydraulic fracture and the gas saturation S.sub.mg in the matrix at different times, thereby obtaining the gas saturation S.sub.mgi,j,t of each matrix grid cell and the gas saturation S.sub.Fgl,t of each hydraulic fracture segment in the target reservoir at time t, so as to calculate the cumulative gas production Q of the gas well produced up to time t by the following equation: $\begin{array}{cc}Q=\underset{i=1}{\overset{n_i}{.Math.}}\underset{{}^{j=1}}{\overset{n_j}{.Math.}}\left(x_{i,j}.Math.y_{i,j}.Math.h\right){\varphi }_m\left(S_{\mathrm{mg}i,j,0}-S_{\mathrm{mg}i,j,t}\right)+\underset{l=1}{\overset{n_l}{.Math.}}\left({\xi }_l.Math.D.Math.h_F\right){\varphi }_F\left(S_{\mathrm{Fg}l,0}-S_{\mathrm{Fg}l,t}\right);& \left(19\right)\end{array}$ where, Q—a cumulative gas production of the gas well from production to time t, in m.sup.3; i, j—Grid coordinates of the matrix; n.sub.i, n.sub.j—a total number of grids in x and y directions in the matrix grid; n.sub.l—a number of the hydraulic fracture segments; S.sub.mgi,j,0—an initial gas saturation in the grid at positions i and j; S.sub.mgi,j,t—a gas saturation in the grid at time t at positions i and j; x.sub.i,j, y.sub.i,j—a length and a width of the grid at positions i and j, in m; S.sub.Fgl,0—an initial gas saturation of the l-th hydraulic fracturing segment; S.sub.Fgl,t—a gas saturation of the l-th hydraulic fracturing segment at time t; ξ.sub.l—a length of the l-th hydraulic fracture segment, in m; h.sub.F—a height of hydraulic fracture, in m; h—a reservoir thickness, in m; if K.sub.F=K.sub.FR, the cumulative gas production Q.sub.1 considering the creep damage is obtained by the above equation; if K.sub.F=K.sub.F0, the cumulative gas production Q.sub.2 without considering the creep damage is obtained by the above equation; the creep damage rate C of the cumulative production is calculated by the following equation: $\begin{array}{cc}C=\frac{.Math.Q_1-Q_2.Math.}{Q_2}\times 100\%;& \left(20\right)\end{array}$ when C≤5%, the creep damage is minimal and the target reservoir is not susceptible to the creep damage; when 5%<C≤10%, the target reservoir is sensitive to the creep damage, and the creep damage is effectively avoided by adjusting fracturing technological parameters; when C>10%, the target reservoir is more sensitive to the creep damage; (5) repeating Steps (3) to (4) and calculating the creep damage rate for cumulative production for the cases of the hydraulic fracture sanding concentration N of 5 kg/m.sup.2, 7.5 kg/m.sup.2, 10 kg/m.sup.2, 12.5 kg/m.sup.2 and 15 kg/m.sup.2 respectively; plotting the creep damage chart of cumulative production, the abscissa represents the sanding concentration, and the ordinate represents the creep damage rate for cumulative production; the creep damage prevention method is identified as follows: if the sanding concentration is 5 kg/m.sup.2 and C≤5%, an on-site fracturing is performed for the well according to recommended parameters; if the sanding concentration N is 5 kg/m.sup.2, 5%<C≤10%, or if N is 5 kg/m.sup.2, C>10% and then ≤10% after N is increased, the creep damage is effectively avoided by adjusting a proppant particle size, a proppant sanding concentration and a flowback regime in the well; if the creep damage rate C of cumulative production is higher than 10% after the sanding concentration N reaches 15 kg/m.sup.2, the creep damage is reduced by increasing the sanding concentration in the well.

2. The method for evaluating and preventing a creep damage to a conductivity of a hydraulic fracture in gas reservoirs according to claim 1, wherein in the step (3), the hydraulic fracture width D is calculated by the following formula: $D=1000\times \frac{N}{\rho };$ where, D—the hydraulic fracture width, in mm; N—the sanding concentration of the proppant, in kg/m.sup.2; ρ—a bulk density of the proppant, in kg/m.sup.3.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0096] FIG. 1 shows the core creep test results and fitting results of Well Z1.

[0097] FIG. 2 shows the core creep test results and fitting results of Well Z2.

[0098] FIG. 3 shows the core creep test results and fitting results of Well Z3.

[0099] FIG. 4 is a chart on creep damage to cumulative production of Wells Z1, Z2 and Z3.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0100] The present invention is further explained according to the drawings and embodiments for those skilled in the art to understand the present invention. It should be understood, however, that the present invention is not limited to the scope of the preferred embodiments. For persons of ordinary skill in the art, these changes are obvious without departing from the spirit and scope of the present invention defined and determined by the appended claims.

Embodiment 1

[0101] Three gas wells (Z1, Z2 and Z3) in a block in northeast Shaanxi were studied as examples to evaluate the creep damage to fracture conductivity of each well with the method proposed in the present innovation and provide associated prevention methods. The following calculations were performed for Wells Z1, Z2 and Z3, respectively.

[0102] Step 1: Selecting rock samples of a target reservoir from three wells to perform rock creep experiments, and plot strain-time curves in the creep process of the three wells.

[0103] Step 2: Fitting the three strain-time curves with the Equations (1) to (3) respectively, obtaining the viscous modulus E.sub.2v as 6,877.57 MPa, 2,294.63 MPa and 3,6751.20 MPa of the experimental rock samples of Wells Z1, Z2 and Z3, as well as the fractional relaxation time τ as 87.22 s.sup.α, 100 s.sup.α and 6.298 s.sup.α and derivative order α as 0.2868, 0.1729 and 0.3766 in the fractional Kelvin model, respectively. The creep experimental data and fitting results for the three wells are plotted as shown in FIGS. 1, 2 and 3.

[0104] Step 3: Substituting the sanding concentration N and bulk density ρ of the proppant for the hydraulic fractures in Wells Z1, Z2 and Z3 after fracturing into the Equation (4) and obtain the widths of hydraulic fractures in the three wells. Next, substituting the obtained parameters into the Equations (5) to (7) and obtain the hydraulic fracture permeability K.sub.FR considering creep damage for the three wells.

[0105] Step 4: Substituting the hydraulic fracture permeability K.sub.FR of the three wells and the reservoir parameters obtained in Step 1 into the Equations (8) to (19) for the three wells, and calculating the cumulative gas production Q.sub.2 of Wells Z1, Z2, and Z3 after 30 days of production considering creep damage. Then, substituting the initial hydraulic fracture permeability K.sub.F0 of the three wells and the reservoir parameters obtained in Step 1 into the Equations (8) to (19), and calculating the cumulative gas production Q.sub.1 of Wells Z1, Z2, and Z3 after 30 days of production without considering creep damage. Next, substituting Q.sub.1 and Q.sub.2 in to the Equation (20) and obtaining the creep damage rate C of cumulative production to determine the creep damage to fracture conductivity of the target reservoir in each well. The creep damage rates C of cumulative production of Wells Z1, Z2 and Z3 were 0.0716, 0.2805 and 0.0155, respectively.

[0106] Therefore, the target reservoir of Well Z1 was sensitive to creep damage which could be effectively avoided by adjusting the fracturing technological parameters. The target reservoir of well Z2 was more sensitive to creep damage and not prone to creep damage.

[0107] Step 5: Repeating Steps 3 to 4 for the cases of sanding concentration N of 5 kg/m.sup.2, 7.5 kg/m.sup.2, 10 kg/m.sup.2, 12.5 kg/m.sup.2 and 15 kg/m.sup.2, and obtaining the creep damage rate C of the cumulative production at different sanding concentrations. Then, plotting the results obtained from the calculations for Wells Z1, Z2 and Z3 onto the creep damage chart of the same cumulative production (FIG. 4).

[0108] According to the creep damage chart, the creep damage rate for cumulative production of Well Z2 was still higher than 10% after the sanding concentration reached 15 kg/m.sup.2. Therefore, the effect of creep damage can only be reduced by increasing the sanding concentration in the well, and the sanding concentration should be increased as much as possible in fracturing operation. Combined with the actual construction conditions, the optimal range of sanding concentration for hydraulic fracturing is 10 kg/m.sup.2 to 12.5 kg/m.sup.2.

[0109] In Well Z1, when the sanding concentration was 5 kg/m.sup.2, C>10% and then <10% after the sanding concentration was increased, creep damage can be effectively avoided by adjusting the proppant particle size, proppant sanding concentration and flowback regime in the well. In construction, the proppant with smaller mesh size should be selected, and the sanding concentration should be controlled at 12 kg/m.sup.2 in combination with the construction conditions. It is also necessary to minimize the contact time between the fracturing fluid and the matrix rock, and slowly increase the flowback speed by forced flowback or flowback method with multiple working systems.

[0110] In Well Z3, when the sanding concentration is 5 kg/m.sup.2 and C<5%, it is unnecessary to consider the damage caused by reservoir rock creep to the conductivity of hydraulic fracture and the on-site fracturing may be continued according to the recommended parameters in development plan.